Shrinkehande graph?

Question

Shrinkhande graph is a graph which is a (16,6,2,2) strongly regular graph. It's well-known that it's a Cayley graph on group $Z_4\times Z_4$ with Cayley set $S=\{ ±(0,1),±(1,1),±(1,0)\}$. I'd like to know if we can define Cayley graph, associated to Shrinkhande graph on group $Z_2^4$. In fact I'm looking for the Cayley set. Besides I know this Cayley graph exists.

Answer 1

There is a list of the 23 Cayley graphs for $\mathbb{Z}^4$ at http://staffhome.ecm.uwa.edu.au/~00013890/. According to sage, in my hands, none of these graphs is isomorphic to the Shrikande graph.

Answer 2

Here is some magma code you can run to find all the regular subgroups of the automorphism group (and thus find all the groups on which it is Cayley):

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G:=DirectProduct(CyclicGroup(4),CyclicGroup(4)); H:=sub< G|G.1,G.2,G.1*G.2>; Gamma:=CayleyGraph(H:Directed:=false,Labelled:=false); A:=AutomorphismGroup(Gamma); for S in Subgroups(A:IsRegular) do IdentifyGroup(S`subgroup); end for;

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You can run this here: http://magma.maths.usyd.edu.au/calc/ and the output is: <16, 2> <16, 11> <16, 6> <16, 6> <16, 8> <16, 8>

We can then check say here: https://people.maths.bris.ac.uk/~matyd/GroupNames/index.html to find out what the groups are: $C_4^2$, $SD_{16}$, $C_2\times D_4$, and $M_4(2)$.